Answer to Question #297149 in Linear Algebra for DaT4

Question

Let A=[a b c         ←Matrix

d e f

g h i]

where a, b, c, d,e, f, g, h, i are some real numbers, if det(A)=5 answer the following questions:

1. Is A invertible? (Justify your answer). Find rank(A)

1. Let b= [a+d b+e c+f         ←Matrix

d      e       f

2g     2h    2i]

And, C= [a b c                   ←Matrix

     -2d -2e -2f

     3g 3h 3I ]

Compute det(B) and det(C).

(C) Compute det(A^-1) and det(adj(A)).

Answer 1

a)

A is invertible because it's determinant is not zero.

Rank of A =3

b)

Det B

Row 2 of A is added to get row 1 of B and twice row 3 of A is row 3 of B.

$\therefore$ det B $=2×5$

$=10$

Det C

(-Twice) row 2 of A and 3 times row 3 of A is row 2 and row 3 of C.

$\therefore$ det C $=(-2)×(3)×(5)$

$=-30$

c)

Det (A) det $($ A$^{-1})=1$

$\implies$ det $(A^{-1})=\frac{1}{5}$

Det (adj A) $=$ det $(A)^{n-1}$

$=(5)^{3-1}$

$=25$